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| [[Image:WeyrMatrixExample.jpg|right|thumb|300px|The image shows an example of a general Weyr matrix consisting of two blocks each of which is a basic Weyr matrix. The basic Weyr matrix in the top-left corner has the structure (4,2,1) and the other one has the structure (2,2,1,1).]]
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| In [[mathematics]], in [[linear algebra]], a '''Weyr canonical form''' (or, '''Weyr form''' or '''Weyr matrix''') is a [[square matrix]] satisfying certain conditions. A square matrix is said to be ''in'' the Weyr [[canonical form]] if the matrix satisfies the conditions defining the Weyr canonical form. The Weyr form was discovered by the [[Czech Republic|Czech]] [[mathematician]] [[Eduard Weyr]] in 1885.<ref>{{cite journal|last=Eduard Weyr|first= |title=Répartition des matrices en espèces et formation de toutes les espèces|journal=[[Comptes Rendus]], Paris|year=1985|volume=100|pages=966–969|url=http://dml.cz/bitstream/handle/10338.dmlcz/400545/DejinyMat_02-1995-1_15.pdf|accessdate=10 December 2013}}</ref><ref>{{cite journal|last=Eduard Weyr|title=Zur Theorie der bilinearen Formen|journal=Monatsh. Math. Physik|year=1980|volume=1|pages=163–236}}</ref><ref name=Weyr>{{cite book|last=Kevin C. Meara, John Clark, Charles I. Vinsonhaler|first= |title=Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form|year=2011|publisher=Oxford University Press}}</ref> The Weyr form did not become popular among mathematicians and it was overshadowed by the closely related, but distinct, canonical form known by the name [[Jordan normal form|Jordan canonical form]].<ref name="Weyr"/> The Weyr form has been rediscovered several times since Weyr’s original discovery in 1885.<ref name="Weyr44">{{cite book|last=Kevin C. Meara, John Clark, Charles I. Vinsonhaler|first= |title=Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form|year=2011|publisher=Oxford University Press|pages=44, 81–82}}</ref> This form has been variously called as ''modified Jordan form,'' ''reordered Jordan form,'' ''second Jordan form,'' and ''H-form''.<ref name="Weyr44"/> The current terminology is credited to Shapiro who introduced it in a paper published in the [[American Mathematical Monthly]] in 1999.<ref name="Weyr44"/><ref>{{cite journal|last=Shapiro, H.|title=The Weyr characteristic|journal=The American Mathematical Monthly|year=1999|volume=106|pages=919–929}}</ref>
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| Recently several applications have been found for the Weyr matrix. Of particular interest is an application of the Weyr matrix in the study of [[phylogenetics|phylogenetic invariant]]s in [[biomathematics]].
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| ==Definitions==
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| ===Basic Weyr matrix===
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| ===Definittion===
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| A basic Weyr matrix with [[eigenvalue]] <math>\lambda</math> is an <math>n\times n</math> matrix <math>W</math> of the following form: There is a [[Partition (number theory)|partition]]
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| : <math>n_1 + n_2+ \cdots +n_r=n</math> of <math>n</math> with <math>n_1\ge n_2\ge \cdots \ge n_r\ge 1</math>
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| such that, when <math>W</math> is viewed as an <math> r \times r</math> [[block matrix|blocked matrix]] <math>(W_{ij})</math>, where the <math> (i, j)</math> block <math> W_{ij}</math> is an <math>n_i \times n_j</math> matrix, the following three features are present:
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| # The main [[diagonal]] blocks <math> W_{ii}</math> are the <math>n_i\times n_i </math> [[scalar matrix|scalar matrices]] <math>\lambda I </math> for <math>i = 1, \ldots , r</math>.
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| # The first [[superdiagonal]] blocks <math>W_{i,i+1} </math> are full [[column rank]] <math>n_i \times n_{i+1}</math> matrices in [[reduced row-echelon form]] (that is, an [[identity matrix]] followed by zero rows) for <math> i=1, \ldots, r-1 </math>.
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| # All other blocks of ''W'' are zero (that is, <math> W_{ij} = 0 </math> when <math>j \ne i, i + 1</math>).
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| In this case, we say that <math>W</math> has Weyr structure <math>(n_1, n_2, \ldots , n_r)</math>.
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| ===Example===
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| The following is an example of a basic Weyr matrix.
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| <center>
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| <math>W = </math>
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| [[File:BasicWeyrMatrix.jpg|A Basic Weyr matrix with structure (4,2,2,1)]]
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| <math> =
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| \begin{bmatrix}
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| W_{11} & W_{12} & & \\
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| & W_{22} & W_{23} & \\
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| & & W_{33} & W_{34} \\
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| & & & W_{44} \\
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| \end{bmatrix}
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| </math>
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| </center>
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| In this matrix, <math> n=10</math> and <math> n_1=4, n_2=2, n_3=2, n_4=1</math>. So <math> W</math> has the Weyr structure <math>(4,2,2,1)</math>. Also,
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| <center>
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| <math>
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| W_{11} =
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| \begin{bmatrix}
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| \lambda & 0 & 0 & 0 \\
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| 0 &\lambda & 0 & 0 \\
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| 0 & 0 & \lambda & 0 \\
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| 0 & 0 & 0 & \lambda \\
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| \end{bmatrix} = \lambda I_4, \quad
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| W_{22} =
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| \begin{bmatrix}
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| \lambda & 0 \\
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| 0 &\lambda & \\
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| \end{bmatrix} = \lambda I_2, \quad
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| W_{33} =
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| \begin{bmatrix}
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| \lambda & 0 \\
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| 0 &\lambda & \\
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| \end{bmatrix} =\lambda I_2, \quad
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| W_{44} =
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| \begin{bmatrix}
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| \lambda \\
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| \end{bmatrix} = \lambda I_1
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| </math>
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| </center>
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| and
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| <center>
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| <math>
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| W_{12}=
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| \begin{bmatrix}
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| 1 & 0 \\
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| 0 & 1\\
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| 0 & 0\\
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| 0 & 0\\
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| \end{bmatrix}, \quad
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| W_{23}=
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| \begin{bmatrix}
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| 1 & 0 \\
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| 0& 1\\
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| \end{bmatrix},\quad
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| W_{34} =
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| \begin{bmatrix}
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| 1 \\
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| 0 \\
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| \end{bmatrix}.
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| </math>
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| </center>
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| ===General Weyr matrix===
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| ===Definition===
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| Let <math> W </math> be a square matrix and let <math>\lambda_1, \ldots, \lambda_k </math> be the distinct eigenvalues of <math>W </math>. We say that <math> W </math> is in Weyr form (or is a Weyr matrix) if <math> W </math> has the following form:
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| <center>
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| <math>
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| W =
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| \begin{bmatrix}
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| W_1 & & & \\
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| & W_2 & & \\
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| & & \ddots & \\
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| & & & W_k \\
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| \end{bmatrix}
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| </math>
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| </center>
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| where <math> W_i </math> is a basic Weyr matrix with eigenvalue <math> \lambda_i </math> for <math> i = 1, \ldots , k</math>.
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| ===Example===
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| The following image shows an example of a general Weyr matrix consisting of three basic Weyr matrix blocks. The basic Weyr matrix in the top-left corner has the structure (4,2,1) with eigenvalue 4, the middle block has structure (2,2,1,1) with eigenvalue -3 and the one in the lower-right corner has the structure (3, 2) with eigenvalue 0.
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| <center>
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| [[Image:WeyrMatrixExample02.jpg]]
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| </center>
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| ==The Weyr form is canonical==
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| That the weyr form is a canonical form of a matrix is a consequence of the following result:<ref name="Weyr"/> ''To within permutation of basic Weyr blocks, each square matrix <math >A</math> over an algebraically closed field is similar to a unique Weyr matrix <math >W</math>. The matrix <math >W</math> is called the Weyr (canonical ) form of <math >A</math>.''
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| ==Computation of the Weyr canonical form==
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| ===Reduction to the nilpotent case===
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| Let <math>A</math> be a square matrix of order <math>n</math> over an [[algebraically closed field]] and let the distinct eigenvalues of <math>A</math> be <math>\lambda_1, \lambda-2, \ldots, \lambda_k</math>. As a consequence of the generalized [[eigenspace]] decomposition theorem, one can show that <math>A</math> is [[matrix similarity|similar]] to a block diagonal matrix of the form
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| <math>
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| A=
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| \begin{bmatrix}
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| \lambda_1I + N_1& & & \\
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| & \lambda_2I + N_2 & & \\
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| & & \ddots & \\
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| & & & \lambda_kI + N_k \\
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \lambda_1I & & & \\
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| & \lambda_2I & & \\
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| & & \ddots & \\
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| & & & \lambda_kI \\
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| \end{bmatrix}
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| +
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| \begin{bmatrix}
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| N_1& & & \\
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| & N_2 & & \\
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| & & \ddots & \\
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| & & & N_k \\
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| \end{bmatrix}
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| =
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| D+N
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| </math>
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| where <math>D</math> is a [[diagonal matrix]] and <math>N</math> is a [[nilpotent matrix]]. So the problem of reducing <math>A</math> to the Weyr form reduces to the problem of reducing the nilpotent matrices <math>N_i</math> to the Weyr form.
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| ===Reduction of a nilpotent matrix to the Weyr form===
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| Given a nilpotent square matrix <math>A</math> of order <math> n</math> over an algebraically closed field <math> F</math>, the following algorithm produces an invertible matrix <math> C </math> and a Weyr matrix <math> W</math> such that <math>W=C^{-1}AC</math>.
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| '''Step 1'''
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| Let <math>A_1=A</math>
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| '''Step 2'''
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| # Compute a [[Basis (linear algebra)|basis]] for the [[null space]] of <math>A_1</math>.
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| # Extend the basis for the null space of <math>A_1</math> to a basis for the <math>n</math>-dimensional vector space <math>F^n</math>.
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| # Form the matrix <math>P_1</math> consisting of these basis vectors.
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| # Compute <math> P_1^{-1}A_1P_1=\begin{bmatrix}0 & B_2 \\ 0 & A_2 \end{bmatrix}</math>. <math>A_2</math> is a square matrix of size <math>n</math> − nullity <math>(A_1)</math>.
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| '''Step 3'''
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| If <math>A_2</math> is nonzero, repeat Step 2 on <math>A_2</math>.
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| # Compute a basis for the null space of <math>A_2</math>.
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| # Extend the basis for the null space of <math>A_2</math> to a basis for the vector space having dimension <math>n</math> − nullity <math>(A_1)</math>.
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| # Form the matrix <math>P_2</math> consisting of these basis vectors.
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| # Compute <math> P_2^{-1}A_2P_2=\begin{bmatrix}0 & B_3 \\ 0 & A_3 \end{bmatrix}</math>. <math>A_2</math> is a square matrix of size <math>n</math> − nullity <math>(A_1)</math> − nullity<math>(A_2)</math>.
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| '''Step 4'''
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| Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices <math>A_1, A_2, A_3, \ldots</math> and associated [[invertible matrix|nvertible matrices]] <math>P_1, P_2, P_3, \ldots</math> until the first zero matrix <math>A_r</math> is obtained.
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| '''Step 5'''
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| The Weyr structure of <math>A</math> is <math>(n_1,n_2, \ldots, n_r)</math> where <math>n_i </math> = nullity<math>(A_i)</math>.
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| '''Step 6'''
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| # Compute the matrix <math> P = P_1 \begin{bmatrix} I & 0 \\ 0 & P_2 \end{bmatrix}\begin{bmatrix} I & 0 \\ 0 & P_3 \end{bmatrix}\cdots \begin{bmatrix} I & 0 \\ 0 & P_r \end{bmatrix}</math> (here the <math>I</math>'s are appropriately sized identity matrices).
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| # Compute <math>X=P^{-1}AP</math>. <math>X</math> is a matrix of the following form:
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| :: <math> X = \begin{bmatrix}0 & X_{12} & X_{13} & \cdots & X_{1,r-1} &X_{1r}\\ & 0 & X_{23} & \cdots & X_{2,r-1} & X_{2r}\\ & & & \ddots & \\ & & & \cdots & 0& X_{r-1,r} \\ & & & & & 0 \end{bmatrix}</math>.
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| '''Step 7'''
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| Use elementary row operations to find an invertible matrix <math> Y_{r-1}</math> of appropriate size such that the product <math>Y_{r-1}X_{r,r-1}</math> is a matrix of the form <math>I_{r,r-1}= \begin{bmatrix} I \\ O \end{bmatrix}</math>.
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| '''Step 8'''
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| Set <math>Q_1= </math> diag <math>(I,I, \ldots, Y_{r-1}^{-1}, I)</math> and compute <math> Q_1^{-1}XQ_1</math>. In this matrix, the <math>(r,r-1)</math>-block is <math>I_{r,r-1}</math>.
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| '''Step 9'''
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| Find a matrix <math>R_1</math> formed as a product of [[elementary matrix|elementary matrices]] such that <math> R_1^{-1} Q_1^{-1}XQ_1R_1</math> is a matrix in which all the blocks above the block <math>I_{r,r-1}</math> contain only <math>0</math>'s.
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| '''Step 10'''
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| Repeat Steps 8 and 9 on column <math> r-1</math> converting <math>(r-1, r-2)</math>-block to <math>I_{r-1,r-2}</math> via [[conjugation (group theory)|conjugation]] by some invertible matrix <math>Q_2</math>. Use this block to clear out the blocks above, via conjugation by a product <math>R_2</math> of elementary matrices.
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| '''Step 11'''
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| Repeat these processes on <math>r-2,r-3,\ldots , 3, 2</math> columns, using conjugations by <math> Q_3, R_3,\ldots , Q_{r-2}, R_{r-2}, Q_{r-1} </math>. The resulting matrix <math>W</math> is now in Weyr form.
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| '''Step 12'''
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| Let <math> C = P_1 \text{diag} (I, P_2) \cdots \text{diag}(I, P_{r-1})Q_1R_1Q_2\cdots R_{r-2}Q_{r-1}</math>. Then <math> W = C^{-1}AC</math>.
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| ==Applications of the Weyr form==
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| Some well-known applications of the Weyr form are listed below:<ref name="Weyr"/>
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| # The Weyr form can be used to simplify the proof of Gerstenhaber’s Theorem which asserts that the subalgebra generated by two commuting <math>n \times n</math> matrices has dimension at most <math>n</math>.
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| # A set of finite matrices is said to be approximately simultaneously diagonalizable if they can be perturbed to simultaneously diagonalizable matrices. The Weyr form is used to prove approximate simultaneous diagonalizability of various classes of matrices. The approximate simultaneous diagonalizability property has applications in the study of phylogenetic invariants in [[biomathematics]].
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| # The Weyr form can be used to simplify the proofs of the irreducibility of the variety of all ''k''-tuples of commuting complex matrices.
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| ==References==
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| {{reflist}}
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| [[Category:Linear algebra]]
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| [[Category:Matrix theory]]
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| [[Category:Matrix normal forms]]
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| [[Category:Matrix decompositions]]
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